Moufang plane

Moufangebenen are projective planes in which the small set of projective Desargues is generally valid. They are named after the German mathematician Ruth Moufang that investigated these levels in the 1930s. You could show that every Moufangebene is isomorphic to a projective plane over an alternative body. All finite Moufangebenen are pappussche levels, all levels are desargueschen Moufangebenen.

Moufangebenen form the class VII in the classification of projective planes by Hanfried Lenz.

Alternatively, a body, it can be made ​​to a projective plane, by using as a projective space to a body generated by an element of the one-dimensional sub-spaces as points, and the two-dimensional sub-spaces as a line. One then speaks of also of the projective plane and listed them as. This projective coordinate planes are always Moufangebenen. Just when the multiplication in the alternative body meets the associative law, a skew field and the plane is a projective plane desarguesian. Note, however, that forms with an alternative body that is not a division ring, none of the formally representable coordinate spaces for a projective geometry, compare to Axiom of Veblen -Young!

Each Moufangebene is isomorphic to a projective coordinate plane via an alternative body, which is uniquely determined by the level up to isomorphism.

With a set of Artin and anger, which states that every finite Alternatively body is a body, it follows that every finite Moufangebene is actually a projective plane over a finite field.

Equivalent descriptions for the term " Moufangebene ": A projective plane is exactly then a Moufangebene when

• Each is an affine translation plane through slots from their resulting affine plane,
• Ternärkörper all, which can be assigned to the level coordinate range by the choice of a projective coordinate system, ie, by selecting a complete tetragon as a point -based are isomorphic
• One of the Koordinatenternärkörper is an alternative body,
• For every line of the plane, the group of collineations which can be laid straight pointwise transitive on the set of points that do not lie on the line, operated,
• The group of collineations transitively on the set of complete quadrilaterals ( construed as an ordered set of four corners) surgery.

At a Moufangebene are said affine translation levels to each other all the isomorphic ( Incidence structures), their Koordinatenternärkörper always quasi body and even Alternatively body, which are likewise each isomorphic.

The real octonions are an example of an alternative body that is not a division ring, the projective plane is the most important example of a nichtdesarguesche Moufangebene.